C₄×D₈×C₃₇ #14557102

order	2368
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₄×D₈×C₃₇

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	D₄
commutator	C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₂×C₄	C₂³×C₃₇
socle	C₂²×C₃₇	C₂×D₄
prime radical	C₂×D₈×C₃₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₄²×C₃₇ ◁ G

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49);(5,6,7,8,9,10,11,12);(6,12)(7,11)(8,10)

Conjugacy classes & character table

1036 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#1331, C₃₇×#1333, C₈×D₈×C₃₇, …
maximal normal subgroup of	C₃₇×#1331, C₃₇×#1333, C₈×D₈×C₃₇, …
maximal quotient of	C₈×D₈×C₃₇, C₃₇×#1581, C₃₇×#1589, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 118);

C4×D8×C37 #14557102